Systems of Equations
Adding and Subtracting Rational Expressions with Different Denominators
Graphing Linear Equations
Raising an Exponential Expression to a Power
Horizontal Line Test
Quadratic Equations
Mixed Numbers and Improper Fractions
Solving Quadratic Equations by Completing the Square
Solving Exponential Equations
Adding and Subtracting Polynomials
Factorizing simple expressions
Identifying Prime and Composite Numbers
Solving Linear Systems of Equations by Graphing
Complex Conjugates
Graphing Compound Inequalities
Simplified Form of a Square Root
Solving Quadratic Equations Using the Square Root Property
Multiplication Property of Radicals
Determining if a Function has an Inverse
Scientific Notation
Degree of a Polynomial
Factoring Polynomials by Grouping
Solving Linear Systems of Equations
Exponential Functions
Factoring Trinomials by Grouping
The Slope of a Line
Simplifying Complex Fractions That Contain Addition or Subtraction
Solving Absolute Value Equations
Solving Right Triangles
Solving Rational Inequalities with a Sign Graph
Domain and Range of a Function
Multiplying Polynomials
Slope of a Line
Multiplying Rational Expressions
Percent of Change
Equations Involving Fractions or Decimals
Simplifying Expressions Containing only Monomials
Solving Inequalities
Quadratic Equations with Imaginary Solutions
Reducing Fractions to Lowest Terms
Prime and Composite Numbers
Dividing with Exponents
Dividing Rational Expressions
Equivalent Fractions
Graphing Quadratic Functions
Linear Equations and Inequalities in One Variable
Notes on the Difference of 2 Squares
Solving Absolute Value Inequalities
Solving Quadratic Equations
Factoring Polynomials Completely
Using Slopes to Graph Lines
Fractions, Decimals and Percents
Solving Systems of Equations by Substitution
Quotient Rule for Radicals
Prime Polynomials
Solving Nonlinear Equations by Substitution
Simplifying Radical Expressions Containing One Term
Factoring a Sum or Difference of Two Cubes
Finding the Least Common Denominator of Rational Expressions
Multiplying Rational Expressions
Expansion of a Product of Binomials
Solving Equations
Exponential Growth
Factoring by Grouping
Solving One-Step Equations Using Models
Solving Quadratic Equations by Factoring
Adding and Subtracting Polynomials
Rationalizing the Denominator
Rounding Off
The Distributive Property
What is a Quadratic Equation
Laws of Exponents and Multiplying Monomials
The Slope of a Line
Factoring Trinomials by Grouping
Multiplying and Dividing Rational Expressions
Solving Linear Inequalities
Multiplication Property of Exponents
Multiplying and Dividing Fractions 3
Dividing Monomials
Multiplying Polynomials
Adding and Subtracting Functions
Dividing Polynomials
Absolute Value and Distance
Multiplication and Division with Mixed Numbers
Factoring a Polynomial by Finding the GCF
Adding and Subtracting Polynomials
The Rectangular Coordinate System
Polar Form of a Complex Number
Exponents and Order of Operations
Graphing Horizontal and Vertical Lines
Invariants Under Rotation
The Addition Method
Solving Linear Inequalities in One Variable
The Pythagorean Theorem
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Factoring simple expressions


Before studying this material you must be familiar with the process of removing brackets. This is because factoring can be thought of as reversing the process of removing brackets. When we factorise an expression it is written as a product of two or more terms, and these will normally involve brackets.

1. Products and Factors

To obtain the product of two numbers they are multiplied together. For example the product of 3 and 4 is 3×4 which equals 12. The numbers which are multiplied together are called factors. We say that 3 and 4 are both factors of 12.


The product of x and y is xy . The product of 5x and 3y is 15xy .


x and 5 y are factors of 10xy since when we multiply x by 5 y we obtain 10 xy .

( x + 1) and ( x + 2) are factors of x + 3 x + 2 because when we multiply ( x +1) by ( x + 2) we obtain x + 3 x + 2.

3 and x -5 are factors of 3 x 15 because 3( x - 5) = 3 x - 15.

2. Common Factors

Sometimes, if we study two expressions to find their factors, we might note that some of the factors are the same. These factors are called common factors .


Consider the numbers 18 and 12.

Both 6 and 3 are factors of 18 because 6 × 3 = 18.

Both 6 and 2 are factors of 12 because 6 × 2 = 12.

So, 18 and 12 share a common factor, namely 6.

In fact 18 and 12 share other common factors. Can you find them?


The number 10 and the expression 15 x share a common factor of 5.

Note that 10 = 5 × 2, and 15 x = 5 × 3 x . Hence 5 is a common factor.


3a and 5a share a common factor of a since

3a = 3 a × a and 5 a = 5 × a . Hence a is a common factor.


8x and 12 x share a common factor of 4 x since

8x = 4x × 2x and 12x = 3x × 4x . Hence 4 x is a common factor.

3. Factoring

To factorize an expression containing two or more terms it is necessary to look for factors which are common to the different terms. Once found, these common factors are written outside a bracketed term. It is ALWAYS possible to check your answers when you factorize by simply removing the brackets again, so you shouldn't get them wrong.


Factorize 15 x + 10.


First we look for any factors which are common to both 15x and 10. The common factor here is 5. So the original expression can be written

15 x + 10 = 5(3x ) + 5(2)

which shows clearly the common factor. This common factor is written outside a bracketed term, the remaining quantities being placed inside the bracket:

15 x + 10 = 5(3 x + 2)

and the expression has been factorized. We say that the factors of 15 x + 10 are 5 and 3 x + 2. Your answer can be checked by showing 5(3 x + 2) = 5(3 x ) + 5(2) = 15 x + 10


Factorize each of the following:

1. 10x + 5y ,

2. 21 + 7x ,

3. xy - 8x ,

4 . 4 x -8xy


1. 5( 2x + y ),

2. 7(3 + x ),

3. x ( y - 8),

4. 4 x (1 - 2y ).

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